Explicit count of integral ideals of an imaginary quadratic field

TL;DR

This paper derives explicit bounds for counting integral ideals in imaginary quadratic fields with a focus on error terms, providing precise asymptotic formulas involving character sums and their bounds.

Contribution

It offers new explicit bounds and asymptotic formulas for counting ideals in imaginary quadratic fields, with detailed error estimates involving character sums.

Findings

Explicit bounds with $O(X^{1/3})$ error term for ideal counts.

Asymptotic formulas for character sum related to non-principal characters modulo 3 and 4.

Quantitative error estimates for sums involving characters in quadratic fields.

Abstract

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the non-principal character modulo $3$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi}{3\sqrt{3}}X +O^*( 1.94\,X^{1/3})$, and that, when $\chi$ is the non-principal character modulo $4$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi }{4}X+ O^*( 1.4\,X^{1/3})$.